%I A086378
%S A086378 1,2,4,6,9,12,16,20,25,30,36,42,49,55,56,64,70,72,81,90,100,110,121,
%T A086378 132,144,153,155,156,169,180,182,196,210,225,240,256,272,289,305,306,
%U A086378 324,342,361,377,380,400,420,441,462,484,504,505,506,529,546,552,576
%N A086378 Sum of successive remainders in computing euclidean algorithm for (1,
               1/sqrt(n)) is rational.
%C A086378 Take (a0,b0) = (1,1/sqrt(n)), then (a1,b1) = (b0,a0-b0*floor(a0/b0)), 
               etc. and call kappa(1/sqrt(n)) = b1+b2+b3+... If kappa(1/sqrt(n)) 
               is rational (which can be easily evaluated thanks to the periodicity 
               of the process for a quadratic number as shown in the following MuPAD 
               program), then n is in the sequence.
%C A086378 An infinity of 2nd degree polynomial functions take all their values 
               over N in the sequence (such as x2, x2+x, 36x2+17x+2, 100x2+150x+55, 
               196x2+97x+12, ...).
%e A086378 7137 is in the sequence because kappa(1/sqrt(7137)) = 7/1098 (in Q).
%o A086378 (MuPAD) kappa_1_over_sqrt := proc(n) local a,b,i,p; begin if (a := sqrt(n)-isqrt(n)) 
               = 0 then return(0) end_if: i := a := simplify(1/a,sqrt); p := 1; 
               b := 0; repeat p := simplify(p*a,sqrt); b := simplify(b*a+a-floor(a),
               sqrt); until (a := simplify(1/(a-floor(a)),sqrt)) = i end_repeat: 
               return(simplify((b/(p-1) + 1/a)/sqrt(n),sqrt)); end_proc:
%Y A086378 Sequence in context: A095114 A062048 A075354 this_sequence A088900 A083392 
               A076921
%Y A086378 Adjacent sequences: A086375 A086376 A086377 this_sequence A086379 A086380 
               A086381
%K A086378 nonn
%O A086378 1,2
%A A086378 Thomas Baruchel (baruchel(AT)users.sourceforge.net), Sep 2 2003